"Like a chicken on a June bug" has the same meaning as "like white on rice"; it's a jocular, (perhaps mostly Southern redneck), non-pejorative way of indicating "acting quickly".

If you'd ever raised what are now called "free-range" chickens, who forage among the weeds in the chicken yard and had seen what happened if a June bug or grasshopper made the error in judgment of landing anywhere near the chickens, you'd see first hand that no time was wasted by what would sooner or later become either a good laying hen or Sunday dinner.